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It sounds like you're talking about a classic example of Incubation.
Incubation is defined as a process of unconscious recombination of thought elements that were stimulated through conscious work at one point in time, resulting in novel ideas at some later point in time.
Here's a great article by John F. Kihlstrom: Intuition, Incubation, and Insight: ...

Check out the book The Psychology of Computer Programming by Gerald Weinberg. Originally published in 1971 it was recently produced in a Silver Anniversary version. I own an original copy and read it again about every five years. Weinberg and Glenford Myers both wrote amazing texts which will be useful forever.
In Weinberg, the situation you are ...

The confusion originates from Sigmund Freud who initialized the field with his idea of the unconscious mind. Freud was of course Austrian, and used the terms das Unbewusste and das Vorbewusste. These are most accurately translated to unconscious and preconscious. The latter is the technical term for what you called 'subconscious'.
The word 'subcoscious' is ...

Intuition, as defined by Wikipedia:
Intuition may be defined as understanding or knowing without conscious
recourse to thought, observation or reason.
Intuition is currently understood to be the subliminal processing of information that is too complex for rational thought, e.g. mate choice. The processes that make up intuition are learned, not ...

The phenomenon's called the incubation effect. Wikipedia operationally defines the incubation effect as any benefit of a break during problem solving. In Wallas’ (1926) four-stage model of innovative problem solving or creativity, the incubation stage is the stage in which one takes some time away from the problem (the stages are: preparation, incubation, ...

Mindfulness meditation can help reduce the intensity of emotional pain. This study explores this further. Even listening to music may help reduce the intensity of pain.
Source
Christopher A. Brown, Anthony K.P. Jones. Meditation experience predicts less negative appraisal of pain: Electrophysiological evidence for the involvement of anticipatory neural ...

What you are describing is a mental bottleneck (or that is the term coined by David Rock Anyway - I actually wrote about it (in the context of Dual N Back training in this blog post - to quote myself
A bottleneck happens when you can't solve a problem because you can't
remove a bad thought from your brain. A bad thought is something you
know ...

What you describe is the textbook definition of a habit: routines of behavior that are repeated regularly and tend to occur subconsciously. These are triggered by some external (say, being behind the driver's well of a car) or internal (say, being upset) stimuli. This activates chunks of procedural memory, which attempts to carry out a task that was usually ...

I believe that this excellent answer can shed some light on why you might be unable to figure out the problem. The answerer described it as mental exhaustion, mental fatigue or mental stress. This, though does not explain why you will suddenly see the solution, but I would guess that the tired part of the brain will continue working on the problem in some ...

Good question and this is very big topic mate. Short answer is - every psychosomatic disorder is unconscious in the sense that it's coming from psychological tension but reflects itself in some unpleasant somatic/bodily reaction (e.g. headache, lower back pain, high blood pressure). In general, even if you are conscious of the source of psychosomatic ...

Annoyingly, it depends what area of the research literature you happen to find yourself in. In the perception literature, the distinction lies between subliminal and supraliminal. In the memory and learning literature, the distinction lies between implicit and explicit. In dual-process decision-making theories, the distinction lies between automatic and ...

This phenomenon is called highway hypnosis (also called driving without attention mode or white line fever) and is an example of procedural memory (or automaticity). Procedural memory is the ability to perform certain tasks without conscious awareness.

A more computational explanation can be found in the expectation-based reasoning literature. The theory suggests that people are always generating expectations of what they expect to sense (see/hear/smell/feel etc) in the near future. These expectations are matched against observations. If expectations and observations match, then all is good. When they ...

Status. If you have a large peer group, you can receive confirmation that the insult is either untrue or irrelevant. You don't need to impress those who are outside your group.
Self-worth. If someone calls you "stupid", yet you've received numerous intellectual awards, it can be rationalized that the comment is untrue.
Acceptance. If you have a flaw, ...

There is nothing surprising or unusual in this. It is a straightforward application of skilled learning (the driving or whatever), habit formation (route following), and having your conscious attention on something else. Back in the 1990s, when I was doing landscaping, I used to make good use of this effect by consciously reviewing things I had recently ...

I'm lucky enough to know a counselor-in-training whose preferred modality is CBT; here's what she suggested.
Identifying those core beliefs is indeed the important part. Useful tools include the CBT Thought Record worksheet, which pretty well explains itself, and the downward arrow technique illustrated in this figure:
(Trader, 2011).
One ...

If you are searching specific part of the brain, I think that frontal regions of cortex will be an answer(In particular, the dorsolateral prefrontal cortex which was associated with self-focused metacognitive evaluation). But, as it common in real life, becoming aware of dreaming state required coordinated work of different parts of brain.
You can read this ...

Probably just a part of the story, but the locus coeruleus is known to be implied in the wake-sleep cycle.
Furthermore, an experiment on cats demonstrated that removing this structure cause actual (but nonsensical) behaviors replacing normal REM sleep phases: cats have no more REM sleep and instead hunt non-existing rats (or something else who knows).
...

Instinct - a motor response initiated by the body totally controlled by an external stimulus. An instinct is by definition, a behavior.
Intuition - A sudden alignment of neuronal pathways that joins weakly associated concepts, ideas, facts, into a neuronal network that pulls conscious selective attention skills away from current activities. May induce a ...

There is some evidence that thought disorder (also called loose association) arises, at least partially, from increased spreading activation; schizophrenics, for example, often show a greater increase in activation to indirectly related words compared to unrelated words, than do non-thought disordered controls. This is primarily a cognitive mechanism, not a ...

First we need to define what "subliminal messaging" means. I take it you mean to send a subliminal command and make someone obey that command.
Based on that definition, here's some of what I find from googling:
Effectiveness
The effectiveness in subliminal messaging has been demonstrated to prime individual responses and stimulate mild emotional ...

In order to find such studies, it's a good idea to start off with mapping out what kind of situations would be relevant - that is, in what cases would it even be possible to communicate with someone's unconsciousnes?
Then you need to define what kind of communication you are interested in.
Here are some ways to find some kind of unconscious communication: ...

It has been a while, but I've always understood the theory to say that the superego, the internalization of values and rules, can cause us to repress ideas that aren't inline with our values. Basically, as we continue to do things that we 'know' are 'wrong' there is a resultant feeling of anxiety, which we often seek to avoid.
There are better ways to ...

Instinct is neither learned nor developmental behavior. Intuition is awareness outside of conscious searching or conscious algorithmic behavior. Autonomous activity is a reflex or heart beating that is automatic (may be a conditioned responding).